A two scale model for liquid phase epitaxy with elasticity

(1)

A Two Scale Model for Liquid Phase Epitaxy

with Elasticity

Von der Fakultät für Mathematik und Physik der Universität Stuttgart zur Erlangung der Würde eines Doktors der

Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung

Vorgelegt von

Michael Kutter

aus Weiden i.d. Opf.

Hauptberichter: Prof. Dr. Christian Rohde

Mitberichter: Prof. Dr. Harald Garcke

Prof. Dr. Anna-Margarete Sändig Tag der mündlichen Prüfung: 09.12.2014

Institut für Angewandte Analysis und Numerische Simulation der Universität Stuttgart 2015

(2)

(3)

Abstract

Epitaxy, a special form of crystal growth, is a technically relevant process for the pro-duction of thin films and layers. It gives the possibility to generate microstructures of different morphologies, such as steps, spirals or pyramids. These microstructures are in-fluenced by elastic effects in the epitaxial layer. There are different epitaxial techniques, one is the so-called liquid phase epitaxy. Thereby, single particles are deposited out of a supersaturated liquid solution on a substrate where they contribute to the growth process.

The thesis studies a two scale model including elasticity, introduced in [Ch. Eck, H. Em-merich. Liquid-phase epitaxy with elasticity. Preprint 197, DFG SPP 1095, 2006]. It consists of a macroscopic Navier-Stokes system and a macroscopic convection-diffusion equation for the transport of matter in the liquid, and a microscopic problem that com-bines a phase field approximation of a Burton-Cabrera-Frank model for the evolution of the epitaxial layer, a Stokes system for the fluid flow near the layer and an elasticity system for the elastic deformation of the solid film. Suitable conditions couple the single parts of the model.

As main result, existence and uniqueness of a solution is proven in suitable function spaces. Furthermore, an iterative solving procedure is proposed, which reflects on the one hand the strategy of the proof of the main result via fixed point arguments and, on the other hand, can be a basis for an numerical algorithm.

(6)

(7)

Zusammenfassung

Epitaxie ist ein technischer Kristallwachstumsprozess zur Herstellung dünner Kristallschichten. Diese weisen dabei oft Mikrostrukturen verschiedener Form und Aus-prägung auf, wie zum Beispiel Stufen, Spiralen oder Pyramiden. Die Ausbildung dieser Mikrostrukturen wird unter anderem durch elastische Effekte in der epitaktischen Schicht beeinflusst. Es gibt verschiedene Varianten der Epitaxie, eine davon ist die sogenannte Flüssigphasenepitaxie. Dabei lagern sich einzelne Teilchen aus einer über-sättigten flüssigen Lösung auf einem Substrat ab und tragen dort zum Wachstum bei. In dieser Arbeit wird ein Zweiskalenmodell zur Flüssigphasenepitaxie unter Berück-sichtigung elastischer Verformungen untersucht. Dieses Modell wurde in [Ch. Eck, H. Emmerich. Liquid-phase epitaxy with elasticity. Preprint 197, DFG SPP 1095, 2006] eingeführt. Es besteht aus makroskopischen Navier-Stokes-Gleichungen und einer makroskopischen Konvektions-Diffusions-Gleichung für den Massentransport in der Flüssigkeit, sowie einem mikroskopischen Problem, zusammengesetzt aus einer Phasen-feldapproximation eines Burton-Cabrera-Frank-Modells zur Beschreibung der epitakti-schen Schicht, einem Stokes System in der Flüssigkeit nahe der Schicht und einer Elastizitätsgleichung für die elastischen Effekte in der Schicht. Die einzelnen Teilprob-leme sind durch geeignete Bedingungen gekoppelt.

Das Hauptresultat der Arbeit ist der Beweis der Existenz und Eindeutigkeit einer Lö-sung des Zweiskalenmodells mit Hilfe von Fixpunktargumenten. Des Weiteren wird eine iterative Lösungsstrategie vorgeschlagen, die auf der einen Seite die Beweisstruktur des Hauptresultats widerspiegelt, und auf der anderen Seite die Grundlage für einen numerischen Lösungsalgorithmus sein kann.

(8)

(9)

Acknowledgement

I would like to remember Prof. Dr. Christof Eck, who passed away far to early in 2011. It was him who provided me the opportunity to work on that topic. The thesis is about his model and his project. Even if he could not see the final results of my work, he influenced many proofs and arguments by his support, especially that which are published in [21]. I hold him in grateful memory.

I would like to thank Prof. Dr. Christian Rohde who made it possible for me to finish this work. To him and to Prof. Dr. Anna-Margarete Sändig, I am very grateful for their support during the last years, in particular the many fruitful hours and hours of discussions. Here, I want to emphasize section 5.3, which results from a close collaboration with Prof. Sändig.

I would like to thank Prof. Dr. Harald Garcke for his support during my visit in Regensburg. These few days have probably been the most effective ones of my whole time as a PhD student. Furthermore, I would like to thank Dr. Julia Kundin and Prof. Dr. Heike Emmerich for the discussions we had in Bayreuth and Stuttgart.

I would like to thank all my colleagues at IANS, the mathematicians, especially Markus H. Redecker and Christian Winkler, as well as the non-mathematicians Ingrid Bock, Brit Steiner and Sylvia Zur.

I would like to thank the German Research Foundation (DFG) for financial support within the project "Homogenisierung bei der Flüssigphasenepitaxie unter Berücksichti-gung elastischer Verformung" (Ec 151/6-1,Ec 151/6-2).

Last but not least, I would like to thank my family: Johanna, Josephine and Malte and Raphael, Jacob, Johannes, Elle, Gisela, Hans-Dieter, Carmen and Rainer.

(10)

(11)

1 Introduction

Many components of electronic devices, think for example of microchips, consist of multiple very thin layers, with thickness of several molecule or atom diameters. It is not possible to obtain such thin films by polishing, so other techniques have to be applied, such as controlled crystal growth. Epitaxy is a special form of crystal growth, where single particles (for example of silicon or a silicon-germanium alloy GexSi1-x) are

deposited on a substrate (made for example of silicon), where they can move on its surface, driven by diffusion, until they leave it again or until they contribute to a growing epitaxial layer. The layer thereby forms monomolecular steps, see Figure 1. In the case of the same material composition in substrate and layer the process is called homoepitaxy, else heteroepitaxy.

Figure 1: Schematic Visualization of a Stepped Epitaxial Layer.

Epitaxy is not the name of one process, but a broader term of several epitaxial tech-niques. Among others, there are molecular beam epitaxy (MBE) and liquid phase epitaxy (LPE), which are frequently used in technical applications. The main difference between these concerns the way of depositing the particles on the growing layer. In MBE, the particles are sent from effusion cells as molecular beams through high vacuum to the substrate, while in LPE they are transported through a supersaturated liquid solution by convection and diffusion.

The main application of epitaxial techniques lies in the production of semiconductor devices like solar cells, integrated circuits, lasers and light emitting diodes. During the growth process, the epitaxial layers usually generate microstructures of differ-ent morphologies such as steps, islands, spirals or pyramids, see Figure 2. These microstructures influence the physical properties of the layer, as for example the electric conductivity, and therefore, it is important to understand how they develop. Different aspects are important thereby, [13], [23], [51], and especially in the case of heteroepitaxy, elastic effects play a significant role. These effects are induced by a so-called misfit between substrate and layer, which occurs due to different crystal structures of the materials of the substrate and the layer.

(12)

12 1 INTRODUCTION

In this thesis, a model for LPE is investigated, where elastic effects are included. Liquid Phase Epitaxy. For a detailed description of the physical principles and the applications of LPE, see [6], [50]. The first model for epitaxial growth goes back to Burton, Cabrera and Frank (BCF model), [8]. It is a semi-discrete model: The diffusion process along the surface is described by continuum equations, while in perpendicular direction, the monomolecular steps are resolved in a discrete way. Originally, the model was developed to describe MBE, but almost all models for LPE base on it, too.

Figure 2: Experimentally observed Pyramidal Microstructure in Ge0.85Si0.15 Growth on

Si(001) from Bi Solution, see [13].

Besides the semi-discrete approach of Burton, Cabrera and Frank, other models have been developed in both directions: On the one hand, there are descriptions of the epitaxial growth process by purely continuum models, [37], [54], [56]. These models describe the height of the solid film by nonlinear partial differential equations and do not resolve the stepped structure of the surface. On the other hand, purely discrete models describe the movement of each particle and their interactions by kinetic Monte Carlo methods, [44]. The major disadvantage of these models is, that they are only applicable at very small length scales.

As a version of the BCF model, phase field models have been established, [17], [31], [36], [40]. Based on the ideas of diffuse interface models for phase transitions in solidification processes, see [9], the steps from one monomolecular layer to another are smoothed, where the thickness of the smooth transition region is controlled by a parameter. In fact, in this context, the edges of the monomolecular steps are considered as phase

(13)

13

transitions, whereby a "phase" does not indicate the state of aggregation (solid/liquid), but the thickness of the epitaxial layer, measured by the number of monomolecular layers. Consequently, not only two but multiple phases are involved in the process. These phase field models, in contrast to "sharp step" BCF models, are easier to handle from the analytical as well as from the numerical point of view.

There are also hybrid models, such as that proposed in [33], where the nucleation of new monomolecular layers is modeled by kinetic Monte Carlo methods, but else, a phase field version of the BCF model is used.

Elastic effects have been considered in a purely continuum model, [56], in a purely discrete model, [44], and in a BCF model, [23].

Some models also include the so-called Ehrlich-Schwoebel barrier, [22], [46], [47]: It describes, that a particle is not as likely to be incorporated, if it approaches a step from above, as if it comes from below. This asymmetry induces an uphill current, see [37]. In [5] and [36], it is explained, how to include that in the BCF context.

In LPE in contrast to MBE, where the deposition of particles is usually modeled by a known deposition rate, a model for the volumic transport of particles in the liquid solution has to be coupled to the model for epitaxial growth. Thereby, not only diffusion but also convection should be included as pointed out in [32].

Figure 3: Simulation of an Epitaxial Layer using a Phase-Field-BCF Model, see [10]. Each of the three types of models (continuum, semi-discrete, discrete) has its advan-tages. Concerning the simulation, kinetic Monte-Carlo models are not able to compute epitaxy on large length scales, while purely continuum models do not fully resolve the microstructure. BCF type models, where phase field approximations suit better for this purpose, are a compromise between both, but nevertheless, the microstructure has to be resolved by a numerical grid, which makes the simulation of a technically relevant devise almost impossible. New possibilities for an efficient implementation are opened up by the application of homogenization techniques, which leads to a formulation of two scale models.

(14)

14 1 INTRODUCTION

Homogenization in LPE. Homogenization, [11], has proven to be a very powerful tool for example in the description of fluid flow in porous media, where in a simple case, Stokes equations have to be solved in a domain with unknown microstructures, [30]. Thereby, the inhomogeneous domain is replaced by a fictive homogeneous one, which possesses approximately the same macroscopic properties. In principle, homogenization methods can also be applied for problems, where the unknown microstructure is of interest and should be computed, see for example [14], [15], [16]. In these cases, it results in a so-called two or multi scale model, consisting of macroscopic equations (with homogenized domains, parameters, et cetera) which are coupled to microscopic cell problems for the determination of the microstructure. Usually, there is one cell problem in each macroscopic point.

For LPE, a two scale model has been derived in [17]. The model describes the transport process in the liquid solution by continuum equations and the epitaxial growth with a phase field version of the BCF model. The homogenization here leads to a macro-scopic domain, which is fully occupied by the liquid solution, and for every point on the substrate, microscopic BCF problems have to be solved for the calculation of the microstructure. Coupling conditions, which act as boundary conditions on the macro-scopic scale, model the interaction between the liquid solution and the epitaxial layer. The well-posedness of the model has been proven and the formal derivation of the two scale model was justified rigorously, see [17].

The model of [17] was further developed in [18] and [19], where elastic effects are included. The main difference to [17] is, that the microscopic cell problems consist not only of BCF models, but also of equations for the description of the elastic effects and the fluid flow near the surface of the layer. The consequences on the mathematical analysis and numerics for the model are tremendous. While the model without elasticity consists essentially of semi-linear partial differential equations, the extended microscopic problems are fully nonlinear.

Objective of the thesis. The goal of the thesis is to analyze the two scale model with elasticity developed in [19]. The focus hereby lies on the proof of existence and uniqueness of a solution, but furthermore, a basis for the numerical treatment is laid. The model consists of macroscopic Navier-Stokes equations and a macroscopic convection-diffusion equation for the transport process in the liquid solution "far away" from the interface between liquid solution and solid layer, and of microscopic cell problems for processes near the interface. These are modeled by Stokes equations for the fluid flow, a linear elastic equation for the deformation in the solid layer and a phase field version of a BCF model for the evolution of the epitaxial surface.

Outline. In chapter 2, the notation is explained. Definitions are given, which are needed throughout the thesis, especially that of some important function spaces. Furthermore, several functional analytical tools are collected from literature which are frequently used in the proofs, and some results are adapted to the cases of the thesis.

(15)

15

model. The original BCF version with sharp steps is explained as well as its phase field approximation. The ansatz for the derivation of the two scale model is given, and the model problem is stated precisely. The chapter concludes with a short discussion of the model.

Chapter 4 proposes an iterative solving procedure with two objectives: On the one hand, it outlines a strategy how existence and uniqueness of a solution of the model can be proven via fixed point arguments, and on the other hand, it is a basis for an algorithm to solve the model problem numerically. The iteration consists of two encapsulated iterations, an inner for the microscopic cell problems and an outer for the coupling between the microscopic and the macroscopic problems. Convergence of the iterative procedure is proven.

The main chapter of the thesis is chapter 5, and it starts by stating the main result in section 5.1: The existence and uniqueness of a solution of the two scale model in suitable function spaces, under appropriate assumptions on the given data. Its proof is given in the following sections. First, the microscopic cell problem is investigated in section 5.2, where the macroscopic coupling data is supposed to be given. Thereto, the single microscopic problems, namely the Stokes problem, the elastic problem and the BCF problem are studied separately in sections 5.2.1 – 5.2.3, and their coupling is investigated in section 5.2.4. Existence and uniqueness of solutions of the coupled microscopic cell problems is proven using Banach’s Fixed Point Theorem. Conversely, solvability of the macroscopic problem is shown in section 5.3 with the microscopic coupling data considered as given. Only the convection- diffusion problem is discussed since the Navier-Stokes equations decouple from the rest of the model. As last step in the proof of the main result, section 5.4 treats the coupling between the microscopic and the macroscopic parts, again by using Banach’s Fixed Point Theorem.

Finally, chapter 6 concludes with a short discussion of results and open problems. Parts of this thesis are to find in [20] and [21] and another publication is in preparation.

(16)

(17)

2 Notation and some Basics

This chapter is devoted to some mathematical fundamentals of this thesis. On the one hand, definitions are provided especially that of several function spaces. On the other hand, some results are given, which are frequently used in chapter 5, such as embedding theorems, basic inequalities, Banach’s Fixed Point Theorem, et cetera. Few of them are not to find in common literature and, in these cases, proofs are presented. The rest is supplied by references.

2.1 Basic Notation

In Rn_{, the i -th standard coordinate vector is denoted by e}

i. For x = (x1, . . . , xn)> ∈ Rn,

|x | is the Euclidean norm of x . If Ω ⊂ Rn_{, then ∂Ω is its boundary, Ω its closure and}

|Ω| its volume.

The scalar product between two vectors x and y in Rn is denoted by x · y , the scalar product between two matrices A = (ai j) and B = (bi j), i = 1, . . . , n, j = 1, . . . , m, in

Rn×m by

A : B = X

i =1,...,n j =1,...,m

ai jbi j.

Consider a time interval I = [0, T ] and a domain Ω_{⊂ R}n_{. For a function f : I}

× Ω → R, the partial derivatives with respect to the time variable t or the space variable xi

are denoted by ∂tf or ∂xif respectively. If β = (β1, . . . , βn)∈ N

n

0 is a multi-index with

|β| = β1+· · · + βn, then the spatial derivatives are expressed by

Dβf := ∂β1 x1 . . . ∂ βn xnf , D k f :={Dβf | |β| = k} and |Dkf| :=   X |β|=k |Dβf|2   1 2 .

For vector valued functions f : I × Ω → _Rm_, _f ₌ _(f

1, . . . , fm), it is

Dβf := (Dβf1, . . . , Dβfm) et cetera.

In the special case of first order derivatives, the gradient of f is defined by

∇f :=    ∂x1f1 · · · ∂x1fm .. . . .. ... ∂xnf1 · · · ∂xnfm   ,

and, in the case n = m, its symmetric part by e(f ) := 1

2 ∇f + (∇f )

>_.

If there is the possibility of confusion, because of the occurrence of different spatial variables x _{∈ R}n_{and y}

∈ Rn_{, the corresponding derivatives are supplied with a respective}

(18)

18 2 NOTATION AND SOME BASICS

2.2 Hölder Spaces

Denote by C∞(Ω) the space of infinitely differentiable functions, defined in a domain Ω_{⊂ R}n. For k _{∈ N}0, let Ck(Ω) be the space of k -times differentiable functions, which

have bounded norm kf kCk_(Ω) = X |β|≤k sup x∈Ω |Dβ_{f (x )|.} Ck

0(Ω) denotes the subset of C

k_{(Ω) of functions with compact support in Ω. For k}

∈ N0

and 0 < α≤ 1, it is

Ck +α(Ω) :={f ∈ Ck(Ω)| Dkf ∈ Cα(Ω)} the classical Hölder space, equipped with the norm

kf kCk +α_(Ω) =kf k_Ck_(Ω)+ max

|β|=k_{x6=y ∈Ω}sup

|Dβ_{f (x )}_{− D}β_{f (y )|}

|x − y |α .

Sometimes, the more compact notation Cθ_{(Ω) with θ = k + α}

∈ R+

0 is used.

In the context of evolution equations, for example parabolic partial differential equations, functions with different regularity properties with respect to time and space are of particular importance.

Definition 2.1 (Some Hölder spaces with anisotropic regularity in time and space). Let be k , ` _{∈ N}0, 0≤ α, α1, α2≤ 1, I = [0, T ] and Ω ⊂ Rn. C0,k +α(I× Ω) = {f ∈ C(I × Ω)| f (t, ·) ∈ Ck +α_(Ω), _{∀t ∈ I},} Ck +α,0(I× Ω) = {f ∈ C(I × Ω)| f (·, x ) ∈ Ck +α(I), ∀x ∈ Ω}, Cα1,α2_(I_{× Ω) = C}α1,0_(I _{× Ω) ∩ C}0,α2_(I _{× Ω),} Ck ,`(I× Ω) = Ck ,0_(I _{× Ω) ∩ C}0,`_(I_{× Ω),} Ck +α1,`+α2_(I_{× Ω) = {f ∈ C}k ,`_(I _{× Ω)| ∂}m t f , D β_f _{∈ C}α1,α2_(I _{× Ω), m ≤ k, |β| ≤ `}.}

The first upper index always denotes the regularity in time, the second that in space.

2.3 Lebesgue and Sobolev spaces

For 1 ≤ r ≤ ∞, Lr(Ω) denotes the Lebesgue space of functions whose r -th power is

integrable (r <∞) or which are essentially bounded (r = ∞), equipped with the norm

kf kLr(Ω) =      Z Ω |f |r_dx 1_r , r <∞, esssup_x∈Ω|f (x )|, r =∞.

(19)

2.3 Lebesgue and Sobolev spaces 19

For k _{∈ N}0 and 1 ≤ r ≤ ∞, it is Wrk(Ω) the Sobolev space of functions whose

derivatives up to order k belong to Lr(Ω), equipped with the norm

kf kWk r(Ω) =                X |β|≤k kDβfkrLr(Ω)   1 r , r < ∞, X |β|≤k kDβ_f_k L∞(Ω), r = ∞.

Additionally, if 0 < α < 1 and s = k + α, W_rs(Ω) is the Sobolev-Slobodeckij space, equipped with the norm

kf kWs r(Ω) =  kf kr_Wk r(Ω)+ X |β|=k Z Ω Z Ω |Dβ_{f (x )}_{− D}β_{f (y )|}r |x − y |n+αr dx dy   1 r .

The Hilbert spaces Ws

2(Ω) are denoted by H

s_{(Ω). W}s r (Ω)

0 _{and H}s_(Ω)0 _{denote the dual}

spaces of W_rs(Ω) and Hs(Ω), respectively. The anisotropic Sobolev spaces Ws1,s2

r (I× Ω) are defined analogously to the anisotropic

Hölder spaces in Definition 2.1. Functions in Ws

r(Ω) are only defined almost everywhere in Ω. Since the boundary ∂Ω

usually is a set of measure zero (with respect to the n-dimensional Lebesgue measure), boundary values of a function f ∈ Ws

r(Ω) are not well-defined in the classical sense.

Therefore, the notion of traces is introduced, see for example [27], chapter 1.5, Theorem 1.5.1.2:

Theorem 2.2 (Traces). Let Ω be a bounded open subset of Rn _{with a C}k +1 _boundary

∂Ω, k an integer ≥ 0. Assume s − 1

r is not an integer, s ≤ k + 1, s − 1

r = ` + σ,

0 < σ < 1, ` an integer≥ 0 and n the outer normal vector. Then the trace operator

tr : f 7→ f ,∂f ∂n, . . . , ∂`_f ∂n` ∂Ω

which is defined for f ∈ Ck +1_{(Ω), has a unique continuous extension as an operator}

from W_rs(Ω) onto ` Y j =0 Ws−j − 1 r r (∂Ω).

This operator has a right continuous inverse which does not depend on r .

The space ˚W_rs(Ω) is the subspace of W_rs(Ω) of functions, which trace on ∂Ω is zero. In the case of an unbounded domain Ω, the subscript loc in W1

r,loc(Ω) denotes the set

of functions, which belong to W1

r ( ˜Ω) for any bounded ˜Ω⊂ Ω with Lipschitz boundary

(20)

Finally, here are some density properties of Sobolev spaces:

Theorem 2.3. Suppose that k _{∈ N}0, 1≤ r < ∞ and that Ω is a bounded domain with

Lipschitz boundary. Then C∞(Ω) is dense in W_rk(Ω).

If Ω has Lipschitz boundary, but is not bounded, then the set of restrictions to Ω of functions in C₀∞_(Rn) is dense in W_rk(Ω).

For the proof, see [1], Theorem 3.17, p.67, and Theorem 3.22, pp.68-70. An obvious consequence is the following lemma:

Lemma 2.4. Suppose k _{∈ N}0, 1≤ r < ∞. Wrk(Ω) is dense in Lr(Ω).

There are also the following density properties of the dual spaces, see [1], 3.14, p.65: Theorem 2.5. Suppose k _{∈ N}0, 1 < r <∞ and 1_r+_r10 = 1. Lr(Ω) is dense in Wrk0(Ω)

0 . Lemma 2.6. Suppose k _{∈ N}0, 1 < r <∞ and 1_r +_r10 = 1. W

k r (Ω) is dense in W k r0(Ω) 0 . Proof. Suppose F ∈ Wk r0(Ω) 0 . Since Lr(Ω) is dense in Wrk0(Ω) 0 , see Theorem 2.5, there is a sequence (Fn)n∈N ⊂ Lr(Ω) with

kFn− F k(Wk r 0(Ω))

0 → 0, for n → ∞.

Since W_rk(Ω) is dense in Lr(Ω), see Lemma 2.4, there is for each n ∈ N a sequence

(fnk)k∈N ⊂ Wrk(Ω) with kFn− fnkkLr(Ω)→ 0, for k → ∞. It follows that kF − fnkk(Wk r 0(Ω)) 0 ≤ kF − Fnk(Wk r 0(Ω)) 0 +kFn− fnkk(Wk r 0(Ω)) 0,

and due to Hölder’s inequality kFn− fnkk(Wk r 0(Ω)) 0 = sup w∈W1 r 0(Ω), kw k W 1 r 0(Ω) =1 Z Ω (Fn− fnk)w dx ≤ kFn− fnkkLr(Ω).

For any ε > 0 and any n _{∈ N, there is a number K}n∈ N such that

kFn− fnkkLr(Ω)<

ε

2, for all k ≥ Kn. Furthermore, there is N _{∈ N such that}

kFn− F k(Wk r 0(Ω)) 0 ≤ ε 2, for all n ≥ N, and consequently kF − fnKnk(Wk r 0(Ω)) 0 ≤ kF − Fnk(Wk r 0(Ω)) 0+kFn− fnKnkLr(Ω)< ε, for all n ≥ N.

(21)

2.4 Periodicity 21

2.4 Periodicity

Periodic functions play an important role in this thesis. Therefore, the notion of peri-odicity is stated precisely, see also [11], [17]:

Definition 2.7 (Y –periodicity). An open, bounded and simply connected domain Y _{⊂ R}n _{with Lipschitz boundary is called periodicity cell, if R}n can be represented as a union of shifted copies of Y with empty intersection:

Rn = [

z∈MY

(z + Y ), (z1+ Y )∩ (z2+ Y ) =∅ for z1 6= z2∈ MY,

where MY ⊂ Rn is a countable set of shifts. A function f defined a.e. in Rn is called

Y –periodic if

f (z1+ y ) = f (z2+ y ) for almost all y ∈ Y and all z1, z2 ∈ MY.

The simplest example for a periodicity cell is Y = [(0, 1)]n and MY = Zn.

Definition 2.8 (Spaces of Y –periodic functions). Suppose Y is a periodicity cell. i) For k _{∈ N}0 and 0 ≤ α ≤ 1, the space Cperk +α(Y ) is the subspace of C

k +α

(Rn_{) of}

Y –periodic functions. The space C_per∞(Y ) is the subspace of C∞_(Rn) of Y –periodic functions.

ii) The space W_r,pers (Y ) with s _{∈ R}+₀ and 1 ≤ r ≤ ∞ is the closure of C∞

per(Y ) with

respect to the Ws

r (Y )–norm.

In chapter 5, there occur microscopic domains, which have the form Y _{× R ⊂ R}3 _(or

are subdomains thereof). The following notation is used: Let be X(Y _{× R) ∈ {C}k +α_(Y

× R), Wk

r (Y × R)}, where k ∈ N0, 0 ≤ α < 1,

1≤ r ≤ ∞. Then denote by Xper(Y × R) ⊂ X(Y × R) the subspace of functions, which

are Y –periodic in X(Ql) with respect to y1 and y2 in the sense of definition 2.8. There

is no periodicity assumption concerning y3.

For anisotropic spaces as e.g. C1,2+α

per (I× Y ), the lower index "per" indicates the

corres-ponding subspace which consists of Y –periodic functions.

2.5 Basic Inequalities

Young’s Inequality Suppose a, b > 0, 1 < r, r0 <∞ and 1 r + 1 r0 = 1. Then ab ≤ a r r + br0 r0 . (2.1) Furthermore, if ε > 0, then ab ≤ εar _{+ c (ε)b}r0_, _(2.2) with c (ε) = 1

(22)

Hölder’s Inequality

Suppose 1≤ r, r0 _{≤ ∞ and} 1 r +

1

r0 = 1. Then for f ∈ Lr(Ω) and g ∈ Lr0(Ω), it is

Z Ω |f g|dx ≤ Z Ω |f |r_dx 1_r Z Ω |g|r0_dx _{r 0}1 . (2.3)

For the proof, see e.g. [25], p.623.

Gronwall’s Inequality

Let f (t) be a nonnegative integrable function on I = [0, T ] which satisfies for a.a. t ∈ I

f (t)≤ c1

Z t

0

f (s) ds + c2

with constants c1, c2 ≥ 0. Then

f (t)≤ c2 1 + c1tec1t

(2.4) for a.a. t ∈ I. In particular, f (t) ≡ 0 a.e. in I, if c2 = 0. For the proof, see e.g. [25],

p.625.

Poincaré’s Inequality

Suppose Ω = [0, d ] _{× R}n−1 with d > 0, then there exists a constant k , which is proportional to d , such that

kf kLr(Ω) ≤ kk∇f kLr(Ω), (2.5)

for all f ∈ W1

r (Ω) with f|x1=0= 0. This can be proven as Theorem 6.30 in [1],

pp.183-184.

2.6 Embeddings and Interpolation

The proofs of section 5.2 use embedding and interpolation theorems several times. Some important ones are stated here.

The following result about the famous Sobolev embeddings can be found e.g. in [27], Ch.1.4.4, pp.27-28.

Theorem 2.9 (Embedding theorem). Suppose Ω has a Lipschitz boundary and s1 ∈ R+0,

s2∈ R with s1 ≥ s2, and 1 < r1, r2<∞ such that s1−_rn

1 = s2−

n

r2. Then the embedding

Ws1

r1(Ω) ,→ W

s2

r2(Ω)

exists and is continuous. If Ω is bounded, the statement is true for s1− _rn

1 ≥ s2−

n r2.

(23)

2.6 Embeddings and Interpolation 23

If s _{∈ R}+_{, 1}

≤ r ≤ ∞, k ∈ N0 and 0 < α < 1 such that s − n_r is not an integer and

s− n

r = k + α. Then the embedding

W_rs(Ω) ,→ Ck +α_(Ω)

exists and is continuous. In the case that s − n

r = k is an integer, the continuous

embedding

W_rs(Ω) ,→ C(k−1)+α(Ω) exists for any 0≤ α < 1.

Next, some interpolatory inclusions in Hölder and Sobolev spaces are provided. They result from interpolation estimates, such as that of the following theorem:

Theorem 2.10 (Interpolation estimate for intermediate Hölder spaces). Suppose that Ω has an uniform Cθ2_{-smooth boundary and that θ}

1, θ2, θ ∈ R+0 with θ1 < θ < θ2. Any

function f ∈ Cθ2_{(Ω) satisfies the estimate}

kf kCθ_(Ω) ≤ ckf k1−λ Cθ1(Ω)kf k λ Cθ2(Ω), with λ = θ−θ1 θ2−θ1.

This follows from [38], Proposition 1.1.3, p.13. For spaces with anisotropic regularity in time and space, a direct consequence of Thm. 2.10 is the following lemma, see [38], Prop. 1.1.4, p.13:

Lemma 2.11. Suppose f ∈ Cα,2α_(I_{×Ω) and define the function ˜}_{f : t} _{7→ ˜}_{f (t) := f (t,}_·).

Then, for any 0≤ β ≤ α ≤ 1

2, it is ˜f ∈ C

α−β_{(I, C}2β_{(Ω)) and satisfies}

k ˜fkCα−β_(I,C2β_(Ω)) ≤ ckf k_Cα,2α_(I×Ω).

f and ˜f in Lemma 2.11 are basically the same functions, only considered from different point of views:

• f is considered as a function depending on x and t with values in R or Rn_.

• ˜f only depends on t, but has values in a function space X, that consists of func-tions Ω_{→ R or Ω → R}n, i.e. ˜f : I → X. In the previous Lemma it is X = C2β_(Ω).

Casually speaking, it is ”f = ˜f ” even if this is formally not correct. In the same sloppy manner, Lemma 2.11 states: "Cα,2α(I × Ω) ,→ Cα−β_{(I, C}2β_{(Ω)) with continuous}

em-bedding". From here on, this thesis will not distinguish between f and ˜f any more in order to simplify the notation.

(24)

Another example for interpolation between time and space regularity is the following, see [38], Lemma 5.1.1, p.176:

Lemma 2.12. Suppose 0≤ α ≤ 1

2. The embedding

C1,2+2α(I× Ω) ,→ Cα(I, C2(Ω)), exists and is continuous.

In Sobolev spaces an analogous result as that of Theorem 2.10 is:

Theorem 2.13 (Interpolation estimate for intermediate Sobolev spaces). Suppose that Ω is bounded with Ck1_{-smooth boundary and that 1 < r}

1, r2 <∞, k1 ≥ k2 ≥ 0, 0 < λ < 1 and k := λk1+ (1− λ)k2, 1_r := _rλ 1+ 1−λ r2 . Any function f ∈ W k1 r1 (Ω)∩ W k2 r2 (Ω) belongs to Wk

r (Ω) and satisfies the estimate

kf kWk r(Ω) ≤ ckf k λ W_r1k1(Ω)kf k 1−λ W_r2k2(Ω).

This follows from [7], Theorem 6.4.5, pp.152-153, see also [14], Theorem 2.2.4, p.23. A consequence for spaces with anisotropic regularity in time and space in this case is Corollary 2.2.6 in [14], p.23. It states:

Lemma 2.14. Suppose that I = [0, T ] and Ω is bounded with Ck-smooth boundary. Then for 0 < r < ∞, k, ` ≥ 0 and 0 < λ < 1, every function f ∈ W`,k

r (I× Ω) belongs

to W_rλ`(I, Wrk (1−λ)(Ω)) and satisfies the estimate

kf k_Wλ` r (I,W k (1−λ) r (Ω)) ≤ kf kW `,k r (I×Ω).

In chapter 5, an interpolatory inclusion is used where the dual space (W1 r0(Ω))

0

is involved. A proof is given here:

Proposition 2.15 (An interpolation estimate between W1

r (Ω) and (W 1 r0(Ω)) 0 ). Suppose 2 ≤ r < ∞, 1 r + 1 r0 = 1 and f ∈ W 1 r (Ω). Then kf kLr(Ω) ≤ ckf k 1 r (W1 r 0(Ω)) 0kf k 1−1 r W1 r(Ω).

Proof. Any f ∈ Lr(Ω) induces an element of (Wr10(Ω))

0 , again denoted by f , by hf , w i := Z Ω f w dx , ∀w ∈ Wr10(Ω), kf k₍ W1 r 0(Ω)) 0 = sup w∈W1 r 0(Ω), kw k_{W 1} r 0(Ω) =1 hf , w i.

Note that, if f ∈ Lr(Ω), then |f |r−1∈ Lr0(Ω) with

kfr−1kL_{r 0}(Ω) = Z Ω |f |(r−1)r0dx _{r 0}1 = Z Ω |f |rdx r−1_r =kf kr−1_L r(Ω),

(25)

2.6 Embeddings and Interpolation 25

since (r− 1)r0 _{= r and} 1 r0 =

r−1

r . If f possesses weak derivatives ∇f , then

∇|f | = sign(f )∇f , where sign(f )(x ) =      1, f (x ) > 0, 0, f (x ) = 0, −1 f (x ) < 0,

see [12], Satz 5.20, p.96. Furthermore, the weak derivatives of f|f |r−2 _{are due to the}

product rule

∇ f |f |r−2_{= |f |}r−2_{∇f + sign(f )(r − 2)f |f |}r−3_{∇f ,}

and it follows with Young’s inequality ∇ f |f |r−2 ≤ c|f | r−2_{|∇f | ≤ c |f |}r−1₊_{|∇f |}r−1_. Therefore, if f ∈ W1 r (Ω), then f|f | r−2_{∈ W}1 r0(Ω) with kf |f |r−2kW1 r 0(Ω) ≤ ckf k r−1 W1 r(Ω).

These preparations lead to kf krLr(Ω) = Z Ω |f |rdx = Z Ω f f|f |r−2_dx =kf |f |r−2_k W1 r 0(Ω) Z Ω f f|f | r−2 kf |f |r−2_k W1 r 0(Ω) dx ≤ kf kr−1 W1 r(Ω)kf k(W_{r 0}1(Ω)) 0

Taking the r -th root on both sides gives the result. Lemma 2.16. Suppose 2 ≤ r < ∞ and f ∈ C1_{(I, (W}1

r0(Ω)) 0 ) ∩ C(I, W1 r (Ω)). Then f ∈ C1r(I, L_r(Ω)) with kf k C1r(I,Lr(Ω)) ≤ c kf kC1_(I,(W1 r 0(Ω)) 0₎+kf k_C(I,W1 r(Ω)) .

Proof. Proposition 2.15 and Young’s inequality imply for t16= t2 ∈ I

kf (t1)− f (t2)kLr(Ω) |t1− t2| 1 r ≤ c kf (t1)− f (t2)k₍_W1 r 0(Ω)) 0 |t1− t2| !1r kf (t1)− f (t2)k 1−1 r W1 r(Ω) ≤ ckf k1r C1_(I,(W1 r 0(Ω)) 0₎kf k 1−1 r C(I,W1 r(Ω)) ≤ ckf kC1_(I,(W1 r 0(Ω)) 0₎+kf k_C(I,W1 r(Ω)) .

(26)

26 2 NOTATION AND SOME BASICS Moreover kf (t)kLr(Ω)≤ kf (t)kWr1(Ω), ∀t ∈ I, and thus kf k C1r(I,Lr(Ω)) = max

t∈I kf (t)kLr(Ω)+ sup_t₁_6=t₂_∈I

kf (t1)− f (t2)kLr(Ω) |t1− t2| 1 r ≤ ckf kC1_(I,(W1 r 0(Ω)) 0₎+kf k_C(I,W1 r(Ω)) .

2.7 Banach’s Fixed Point and Lax-Milgram’s Theorem

Both theorems are used frequently in chapter 5 and are therefore provided here: Theorem 2.17 (Lax-Milgram). Suppose H is a real Hilbert space and

a : H_{× H → R}

a bilinear form that satisfies for all u, v ∈ H

a(u, v ) ≤ c1kukHkv kH, (Continuity)

a(u, u) ≥ c2kuk2H, (H-Ellipticity)

with constants c1, c2 > 0. Then, for any element ` of the dual space H0, there exists a

unique element u ∈ H such that a(u, v ) = hf , v i, ∀v ∈ H.

Furthermore, there exists a constant c > 0 such that kukH ≤ ck`kH0.

For a proof, see for example [43], Theorem 9.14, pp.290-292.

Theorem 2.18 (Banach’s Fixed Point Theorem). Suppose M ⊆ X is a non-empty closed subset of a complete metric space (X, d ) and T is an operator

T : M ⊂ X → M,

which is a strict contraction, that means that there is a number 0 ≤ k < 1 such that d (T x , T y )≤ kd (x , y ), ∀x , y ∈ M.

Then, T has a unique fixed point x ∈ M. Furthermore, the sequence (xn)n∈N0, defined

by

x0 ∈ M, xn+1 := T xn, n∈ N0,

converges to x for any x0 ∈ M, and satisfies the a priori estimate

d (xn, x )≤

kn

1− kd (x0, x1).

(27)

2.8 From the Theory of Semigroups 27

2.8 From the Theory of Semigroups

Some tools from semigroup-theory are used in section 5.3. In view of the proofs there, some basic terms are provided here. The following definitions coincide with the termi-nology in [38]. Further information can be found there, another textbook on that topic is for example [41].

Suppose X is a Banach space. The space of bounded linear functionals X → X is denoted by L(X). Consider a linear operator

A : D(A)⊂ X → X,

with domain D(A). The resolvent set ρ(A) is defined by

ρ(A) :=_{{λ ∈ C | (λ I −A)}−1 exists and belongs to L(X)}. The spectrum of A is ς(A) := C \ ρ(A). For λ ∈ ρ(A), the operator

R(λ, A) := (λ I−A)−1: X → X

is called resolvent operator or simply resolvent. The operator A is said to be sectorial, if there are constants ω _{∈ R,} π₂ < θ < π and M > 0 such that

(i ) ρ(A)⊃ Sθ,ω :={λ ∈ C | λ 6= ω, | arg(λ − ω)| < θ},

(i i ) kR(λ, A)kL(X)≤

M

|λ − ω|, ∀λ ∈ Sθ,ω.

In fact, there is a useful condition on how to check, if a linear operator A is sectorial. It is given by the following proposition, which can be found in [38], Proposition 2.1.11, p.43:

Proposition 2.19. Suppose A : D(A) ⊂ X → X is a linear operator such that ρ(A) contains a half plane_{{λ ∈ C | Re λ ≥ ω}, and}

kλR(λ, A)kL(X)≤ M, if Re λ≥ ω,

with ω_{∈ R and M > 0. Then, A is sectorial.}

If A is a sectorial operator, set e0A_{x := x for all x} _{∈ X and define for t > 0 the linear}

bounded operator etA _{by the Dunford integral}

etA := 1 2πi

Z

ω+γr,η

etλR(λ, A)dλ,

where r > 0, π₂ < η < θ and γr,η is the curve {λ ∈ C | | arg λ| = η, |λ| ≥ r }∪

{λ ∈ C | | arg λ| ≤ η, |λ| = r }, oriented counterclockwise.

The family {etA_{| t ≥ 0} of bounded linear operators is said to be the analytic}

(28)

In fact, it can be shown, that λ 7→ eλt_{R(λ, A) is holomorphic in S}

θ,ω and thus, the

definition of etA is independent of r and η. Furthermore, the mapping t 7→ etA _is

analytic from (0,∞) to L(X) and it satisfies the semigroup property etAesA = e(t+s)A, ∀s, t ≥ 0,

see [38], chapter 2, for more information and proofs. Finally, it is lim

t→0e

tA_{x = x} _⇐⇒ _x _{∈ D(A).}

If D(A) is dense in X, this is satisfied for all x ∈ X. Then {etA_{| t ≥ 0} is said to be}

(29)

3 Modeling Liquid Phase Epitaxy: Physical Model and

Two Scale Model

There are several approaches to model liquid phase epitaxy (LPE), as mentioned in the introduction. This thesis analyses a two scale model, proposed in [19]. In this chapter, the model is introduced. First, the non-homogenized model is presented in sections 3.1 and 3.2, and second, the two scale model in sections 3.3 and 3.4. The latter is derived from the non-homogenized model using homogenization techniques. The ansatz for the derivation is explained in section 3.3, but for technical details, the reader is referred to [19].

3.1 Physical Model

The physical situation is the following: Consider a time interval I = [0, T ] and a domain Q _{⊂ R}3 _{which has the form of a container, see Figure 4, and is filled with a liquid}

solution that contains the particles from which an epitaxial layer grows on a substrate, compare also Figures 2 and 3. The contact to the substrate is at the bottom of Q, which is denoted by

S0:={x ∈ Q | x3 = 0}.

The solid film grows on S0, the time dependent domain occupied by that film is denoted

by QS _{= Q}S_{(t). The liquid domain is Q}L_{(t) = Q}_{\ Q}S_(t).

QL QS Liquid Solution Epitaxial Layer S S0

Figure 4: Liquid Phase Epitaxy.

It is assumed that the interface S between QS _{(solid material) and Q}L _{(liquid solution)}

can be represented as the graph of a function h : S0→ [0, ∞) over S0:

S(t) ={x ∈ Q | x3 = h(x1, x2, t)},

and therefore

(30)

30 3 MODELING LIQUID PHASE EPITAXY

The process is modeled as step by step growth, see Figure 5(b). This means, that the solid film grows one monomolecular layer after another. The description of the steps can be reduced to a two dimensional problem by considering a step as a curve in the two dimensional domain S0, see Figure 5(a). The union of these curves is denoted by

Λ = Λ(t).

x1

x2

Step Curves Λ

(a) Top View.

x1

x3

Liquid Solution Solid Layer Steps

(b) Cross Section.

Figure 5: Step by Step Growth.

Thus, the model contains two different types of free boundaries: One is the the interface S = S(t) between the liquid solution and solid layer, the other are the step curves Λ = Λ(t) which are introduced above.

The processes to describe are, see Figure 6:

i) Volumic transport of the particles in the liquid solution, driven by convection and diffusion.

ii) Adsorption of particles to the surface. At that stage, the particles are called adatoms.

iii) Surface diffusion: Adatoms move on the surface, driven by diffusion, until they desorb into the liquid solution or they reach a step and incorporate into the layer. iv) Elastic effects in the layer.

Summarizing, there are three different types of processes: in the liquid, in the solid and on the interface. In order to model these effects, partial differential equations are formulated in each of these three parts and suitable coupling conditions are derived. Hereby, the coupling takes place at the interface S. More precisely, the model is composed of the following parts, for further explanation see [19]:

(31)

3.1 Physical Model 31

Surface Diffusion

Incorporation Adsorption

Desorption

Figure 6: Processes in LPE.

• In I × QL_{, a Navier-Stokes system has to be solved for the fluid flow and the}

pressure, and a convection-diffusion equation for the transport of particles in the liquid solution,

div v = 0, ∂tv + (v · ∇)v − η∆v + ∇p = 0,

(3.1)

∂tcV+ v · ∇cV − DV∆cV = 0, (3.2)

where v is the fluid velocity, p the pressure, cV the mass specific volume concen-tration of particles in the liquid solution ("V" stands for "volume"), η the viscosity of the liquid and DV the diffusion constant of the volumic diffusion process. Boundary conditions are

v = 0, on ∂QL\ S, (3.3) v = J_S−1 1 %V − 1 %E cV τV − cs τs n, on S, (3.4) DV∂c V ∂n = 0, on ∂Q L \ S, (3.5) DV∂c V ∂n = J −1 S (1− c V₎ cs τs − c V τV , on S, (3.6)

where cs is the surface concentration of adatoms (see also the BCF-model for the

evolution of the interface), JS =p1 + |∇h|2is the density of the surface measure

of S, parameterized over S0, %V and %E are the densities of the liquid solution and

the solid layer respectively, τs and τV describe the rates of adsorption and

desorp-tion of adatoms from and to the liquid soludesorp-tion, and n = √ 1

1+|∇h|2((∇h)

>_,₋₁₎>

is the outer normal on ∂QL at S. The coupling conditions (3.4) and (3.6) are derived under the assumption of conservation of the total mass and of the mass of adatoms, see [19], pp.4-5.

(32)

Furthermore, there are initial conditions

v (·, 0) = vi ni, cV(·, 0) = ci niV . (3.7)

• For the description of the elastic effects, there is for each t ∈ I a quasi-stationary elasticity equation

− div σ(u) = 0, in QS, (3.8)

with displacement field u, stress tensor σ(u), given by linear Hooke’s law σ(u) = Ce(u) with linearized strain tensor e(u) = 1₂(∇u + (∇u)>_{) and elastic}

material tensor C. The elastic deformation is driven by a misfit between substrate and epitaxial layer, which occurs due to different crystal structures. A simple model therefor is a prescribed misfit displacement b, that leads to the boundary condition

u = b, on S0. (3.9)

The coupling to the liquid solution is derived from the equilibrium of normal stresses

σ(u)n− η e(v ) n + pn = 0, on S, (3.10)

where n = √ 1

1+|∇h|2(−(∇h)

>_{, 1)}>_{is the outer normal on ∂Q}S _{at S. The boundary}

condition

σ(u)n = 0, on ∂QS\ (S0∪ S), (3.11)

completes this part of the model.

• The evolution of the epitaxial layer is described by a Burton-Cabrera-Frank (BCF) model ∂tcs = Ds∆cs + cV τV − cs τs , t ∈ I, x ∈ S0\ Λ(t), (3.12) cs = ceq 1 + κγ %sRT + hA 2RT σ(u) : e(u), t ∈ I, x ∈ Λ(t), (3.13) vΛ = Ds %s ∂cs ∂n , t ∈ I, x ∈ Λ(t). (3.14)

Here cs is the surface concentration of adatoms ("s" stands for "surface"),

mea-sured by the mass of adatoms per unit area, %s = mA

AA with mass mA and area

AA of one adatom is the surface density of adatoms, Ds the surface diffusion

constant, ceq the equilibrium surface concentration at the monomolecular step,

κ the curvature of the step, γ the step stiffness, R = kB

mA the gas constant with

(33)

3.2 Phase Field Approximation 33

the velocity of the steps. The bracket ∂cs

∂n denotes the difference of the normal

derivatives on both sides of the curves Λ, ∂cs ∂n =∇c+ s · n +₊_∇c− s · n −_,

where c_s± = limr→0cs(x + r n±) for x ∈ Λ with normal vector n+ = n on Λ and

n− =−n. This part of the model is formulated on the surface S0 and, therefore,

the spatial derivatives have to be understood as two dimensional (with respect to x1 and x2).

Finally, there is the boundary condition Ds

∂cs

∂n = 0, on I× ∂S0, (3.15)

and initial conditions

cs(·, 0) = cs,i ni, Λ(0) = Λi ni. (3.16)

• The evolution of the interface S is described by ∂th = 1 %E cV τV − cs τs , (3.17) with h(·, 0) = hi ni.

In contradiction to the concept of step by step growth, with sharp step edges, the interface S is considered as smooth surface in the context of the fluid flow and elasticity problems. The authors in [19] justify this by the fact, that the equations there are continuum equations and that their scale is much larger than that of the monomolecular layers.

Furthermore, for the analysis as well as for the numerics, a smooth transition from step to step is more convenient than the modeling by sharp steps. An approach therefor is the formulation of a phase field approximation of the BCF model, which is presented in the next section.

3.2 Phase Field Approximation

Introduce a phase field function φ : S0 → [0, ∞) which describes the height of the

epitaxial film over a point on S0 by the number of monomolecular layers, see Figure 7.

The use of the notion of "phase" and "phase field" expresses the mathematical similarity of what follows to diffuse interface models for solidifaction processes, see [9]. But a "phase" here does not indicate the state of aggregation (solid/liquid), but the thickness of the epitaxial layer, measured by the number of monomolecular layers, and a step is interpreted as a phase transition. So, multiple phases occur in the process. The natural

(34)

φ = 1 φ = 2 φ = 3 φ = 2

Liquid Solution Solid Layer

Figure 7: The Phase Field.

values of φ would be the nonnegative integers, but φ is allowed to take on real values in a neighborhood of a step which enables a smooth transition from step to step. The BCF model (3.12) - (3.16) is replaced by a phase field approximation. It is derived from the free energy functional

F (φ) = Z S0 hc_eqγβ %s ξ 2|∇φ| 2₊ 1 ξf (φ) − RT (cs − ceq) g1(φ) + hA 2 σ(u) : e(u) g2(φ) i dx ,

with a multi-well potential f which has its minima at integer values, for example f (φ) =− cos(2πφ). The parameter ξ describes the thickness of the smooth transition regions. In [19] the functions g1 and g2 are chosen as g1(φ) = g2(φ) = φ. Following

the suggestions of [31] (where a model without elastic effects is discussed), another possible choice is g1(φ) = 1 2 φ− sin(2πφ) 2π , (3.18)

which keeps the minima of the corresponding term in F with respect to φ at integer values φ_{∈ N}0. Another possible choice for g2 is discussed at the end of this chapter.

The parameter β is given by β−1 =

Z +∞

−∞

(ϕ0(x ))2+ f (ϕ(x )) dx,

where ϕ, which determines the shape of the diffuse transition region at a step, is the solution of −ϕ00(x ) + f0(ϕ(x )) = 0, lim x→−∞ϕ(x ) = 0, x→+∞lim ϕ(x ) = 1, ϕ(0) = 1 2. As in [9], the ansatz α∂tφ =−DφF (φ),

(35)

3.3 Ansatz for the Derivation of the Two Scale Model 35

where α is a relaxation parameter and DφF the Gâteaux derivative of F with respect

to φ, leads to α∂tφ = ceqγβ %s ξ∆φ− 1 ξf 0_(φ) + RT (cs − ceq) g10(φ)− hA 2 σ(u) : e(u) g 0 2(φ)

in I× S0. After rescaling, this results in the phase field equation

τ ξ2∂tφ− ξ2∆φ + f0(φ) + q(φ, cs, u) = 0, in I× S0, (3.19) with τ = α%s ceqγβ and q(φ, cs, u) = ξRT %s ceqγβ (ceq − cs) g10(φ) + ξhA%s 2ceqγβ σ(u) : e(u) g₂0(φ). (3.20)

With the choice of [19], the functions g₁0 and g₂0 are constant g₁0(φ) = g₂0(φ) = 1, while from (3.18), the derivative g₁0 acts like a switch: The corresponding term is only nonzero, if φ /_{∈ N}0, which is only in the transition regions in the neighborhood of a step.

For the analysis in chapter 5, both choices are allowed. φ is endowed with an initial condition

φ(·, 0) = φi ni, (3.21)

and a boundary condition ∂φ

∂n = 0, on I× ∂S0. (3.22)

The surface diffusion equation (3.12) is modified to

∂tcs+ %s∂tφ− Ds∆cs = CV τV − cs τs , (3.23)

compare the corresponding equation in [9]. The additional term %s∂tφ describes the

conservation of adatoms.

The BCF model (3.12) - (3.14) can be interpreted as a sharp interface limit of (3.19), (3.23), see [17], [24], [31].

3.3 Ansatz for the Derivation of the Two Scale Model

The single processes during the growth of the epitaxial layer have completely different length scales. The smallest is that of a particle diameter, which is approximately the height hA of one monomolecular layer, the largest is that of the continuum equations

for the fluid flow and the typical size of the microstructure lies somewhere in between. The main idea of the two scale formulation of the model is, to use different spatial variables for processes with different length scales. The model is derived by homoge-nization techniques with homogehomoge-nization parameter ε. Here, ε represents the scale of

(36)

ε

S0 Y

Figure 8: Periodic Homogenization in the x1-x2-Plane.

the microstructure. In the following, the ansatz is explained and the resulting model is presented. For the technical details, see [19].

As ansatz it is assumed that the quantities in the physical model can be written as power series with respect to ε. Thereby, two different concepts are applied for different space directions:

In x3-direction the existence of a fictive boundary layer is assumed, see Figure 9. For

the velocity field v , the pressure p and the volume concentration cV there are outer expansions , which are valid "far away" from the interface (far field), inner expansions for the boundary layer (near field) and matching conditions between them. The outer expansions are

vε(x , t) = V0(x , t) + εV1(x , t) + . . . ,

pε(x , t) = P0(x , t) + εP1(x , t) + . . . ,

c_εV(x , t) = εC₀V(x , t) + ε2C₁V(x , t) + . . . ,

where the lower index ε indicates the problem of scale ε.

The inner expansions are coupled with periodic homogenization in the x1-x2-plane: It

is assumed that the epitaxial layer forms an approximately periodic microstructure, see Figure 8. Therefor, asymptotic expansions for oscillations on the small scale ε are assumed. This affects the elastic displacement field, the quantities of the BCF-model and the inner expansions of the fluid flow and the volumic transport process, but not their outer expansions. A microscopic space variable y _{∈ Y × R}+ _{is introduced, where}

Y is a two dimensional periodicity cell, in the simplest case Y = [(0, 1)]2. The limit y3→ ∞ has to be interpreted as the "border" between near and far field.

In the asymptotic expansions, the variable y is set to y = x_ε. The inner expansions in the fluid are

vε(x , t) = v0(x , t,x_ε) + εv1(x , t,x_ε) + . . . ,

pε(x , t) = p0(x , t,x_ε) + εp1(x , t,x_ε) + . . . ,

(37)

Far Field

Near Field

Matching of Far and Near Field

Figure 9: Inner and Outer Expansions.

For the elastic displacement, it is

uε(x , t) = εu0(x , t,x_ε) + ε2u1(x , t,x_ε) + . . . ,

for the surface concentration

cs,ε(x , t) = εcs,0(x , t,x_ε) + ε2cs,1(x , t,x_ε) + . . . ,

and the height of the interface S hε(x , t) = εh0(x , t,x_ε) + ε

2

h1(x , t,x_ε) + . . . .

For the step boundaries Λε at scale ε, it is assumed that

Λε(t) = S0∩

[

z∈MY

ε (z + Λε(εz , t)) ,

where MY is the set of shifts, see Definition 2.7. The surfaces Λε(x , t) ⊂ Y depend in

a suitable continuous sense on t, x and ε. Furthermore is supposed, that there is some surface Λ0_{(x , t) and a function π}ε_{(x , t,}_{·) : Λ}0_{(x , t)} _{→ Y , such that Λ}ε_{(x , t) converge}

to Λ0_{(x , t) in some sense as ε}_{→ 0, and}

Λε(x , t) ={πε_{(x , t, y )}_{| y ∈ Λ}0_{(x , t)}.}

Finally, for the velocity and curvature of the steps, it is assumed that vΛε= εvΛ0+ ε2vΛ1+ . . . ,

κε= ε−1κ0+ ε0κ1+ . . . .

The quantities vΛε and κε are evaluated at (x , t, πε(x , t, y )), while vΛi and κi are

eval-uated at (x , t, y ), with y ∈ Λ0_{(x , t).}

The above expansions already propose a scaling for the respective quantities. Further-more, it is necessary to scale material parameters and given data in the inner expansions. The parameter scalings are

(38)

The scale of all other parameters is ∼ 1. As initial conditions is assumed that vε(x , 0) = vi ni(x ), cεV(x , 0) = εc

V

i ni(x ), cs,ε(x , 0) = εcs,i ni(x ,x_ε), x ∈ S0,

with a given functions vi ni, ci niV and cs,i ni, where cs,i ni(x , y ) is Y -periodic with respect to

the second variable. For the boundary condition for u on S0, it is supposed that

bε(x ) = εb0(x ,x_ε),

where b0(x , y ) also is Y -periodic with respect to the second variable.

The "homogenized" domain for the macroscopic space variable x is Q, see Figure 10, and for any x ∈ S0 there is a microscopic domain Y × R+, which consist of the solid

part

QS_Y(x , t) :=_{{y ∈ Y × R}+| y3 < h0(x , t, y1, y2)},

the liquid part

QL_Y(x , t) :=_{{y ∈ Y × R}+| y3 > h0(x , t, y1, y2)},

and the interface

SY(x , t) :={y ∈ Y × R+| y3= h0(x , t, y1, y2)}, Q QL Y QS Y SY x ∈ S0

Figure 10: Macroscopic and Microscopic Domains.

The derivation of the two scale model works out as follows: Insert the above expansions into the model equations of section 3.1 and order by powers of ε. For most parts of the model, only the problems for the lowest order of ε are considered, see [19] for the precise derivation. The results are:

(39)

The Equations in the Liquid Solution

For the lowest order of the outer expansions, the resulting system of differential equa-tions is divxV0= 0, ∂tV0+ (V0· ∇x)V0− η∆xV0+∇xP0= 0, ∂tC0V + V0· ∇xC0V − D V_∆ xC0V = 0, in I × Q.

For the inner expansions the lowest order term of the velocity vanishes v0≡ 0,

and the lowest order term of the volume concentration is constant with respect to y . The matching between inner and outer expansions then leads to

V0(x , t) = 0, c₀V(x , t, y ) = C₀V(x , t), on I× S0, and further to DV∂C V 0 ∂n = ¯cs,0 τs − C V 0 τV , on I× S0, where n = (0, 0,−1)> _{and ¯}_c s,0(x , t) = R

Y cs,0(x , t, y ) dy is the microscopic mean value

of cs,0. For the pressure it is

lim

y3→∞

p0= P0.

In view of the elasticity problem, also the next order term v1 of the fluid velocity is

needed, because it occurs there in a boundary condition. The corresponding equations for (x , t)∈ S0× I are

divyv1= 0,

−η∆yv1+∇yp0= 0,

in QL_Y(x , t),

with the following condition at the interface SY

v1= Js−1 1 %V − 1 %E CV 0 τV − cs,0 τs n, on SY(x , t), (3.24)

where n is the outer normal on QL

Y at SY. The matching to the outer expansion for v1

leads to lim

y3→∞

(40)

The Equations in the Solid Domain

The leading order term solves for any (x , t)∈ S0× I the equation

− divyσ(u0) = 0, in QSY(x , t).

The boundary condition at SY(x , t) is

σy(u0)n− ηey(v1)n + p0n = 0, on SY(x , t).

The Equations of the BCF Model

The lowest order terms solve for x ∈ S0 the equations

∂tcs,0 = Ds,0∆ycs,0+ C₀V τV − cs,0 τs , in I × Y \ Λ0_{(x , t),} cs,0 = ceq,0 1 + κ0γ0 %s,0RT + hA,0 2RT σy(u0) : ey(u0), on I× Λ 0_{(x , t),} vΛ0= Ds,0 %s,0 ∂cs,0 ∂n , on I× Λ0_{(x , t).}

The Phase Field Approximation of the BCF Model

Analogously to the explanations of section 3.2, also the two scale version of the BCF model can be replaced by a phase field approximation. It results in

τ0ξ20∂tφ0− ξ20∆yφ0+ f0(φ0) + q0(φ0, cs,0, u0) = 0, ∂tcs,0+ %s,0∂tφ0− Ds,0∆ycs,0 = C₀V τV − cs,0 τs , in I × Y, with q0(φ0, cs,0, u0) = ξ0RT %s,0 ceq,0γ0β (ceq,0− cs,0)g10(φ0) + ξ0hA,0%s,0 2ceq,0γ0β σy(u0) : ey(u0),

and the scaled parameters ξ = εξ0 and τ = ε−2τ0.

Use of the Phase Field for the description of the interface

The interpretation of the phase field is as already explained above: It denotes the number of monomolecular layers over a point y ∈ Y . This allows to replace the height function h0for the description of the interface between solid layer and liquid solution. The thesis

uses this approach and the microscopic domains are modified correspondingly, see Figure 11. That means that QL

Y is replaced by

(41)

3.4 The Two Scale Model 41 QS Y by Qs(x , t) ={y ∈ R3| (y1, y2)∈ Y, 0 < y3 < hA,0φ0(t, x , y1, y2)}, and SY by Γ(x , t) = _{{y ∈ R}3| (y1, y2)∈ Y, y3 = hA,0φ0(t, x , y1, y2)}.

As simplification, the boundary condition (3.24) is replaced by

v1=− 1 %V − 1 %E CV 0 τV − cs,0 τs e3, on Γ.

This only is a modification at the steps. In fact it seems to be feasible from the physical point of view: The normal direction on Γ at a step is approximately perpendicular to the x3-direction and since the height of a step is of the size of one particle diameter,

there should not be a normal velocity of the fluid in that direction.

Concluding this chapter, the two scale model to be analyzed with completing boundary and initial conditions is summarized in the next section.

3.4 The Two Scale Model

The rest of this thesis is concerned with the following two scale model. In order to simplify the notation, the indices of the asymptotic expansions are omitted:

v (x , t, y ) = v1(x , t, y ) microscopic fluid velocity

V (x , t) = V0(t, x ) macroscopic fluid velocity

p(x , t, y ) = p0(x , t, y ) microscopic pressure

P (x , t) = P0(x , t) macroscopic pressure

CV(x , t) = C₀V(x , t) volume concentration of particles in the liquid solution

φ(x , t, y ) = φ0(x , t, y ) phase field

cs(x , t, y ) = cs,0(x , t, y ) surface concentration of adatoms

u(x , t, y ) = u0(x , t, y ) elastic displacement

Capital letters denote purely macroscopic quantities, small letters indicate quantities depending on x and y . Also the index "0" for the parameters and the given data is omitted.

(42)

Ql

Qs

Γ =_{{y ∈ Y × R}+| y = (y1, y2, hAφ(y1, y2))>}

Y × {0}

Figure 11: The Microscopic Interface and the Phase Field.

The model is composed of:

• Macroscopic Navier-Stokes equations and a convection-diffusion equation in I ×Q divxV = 0,

∂tV + (V · ∇x)V − η∆xV +∇xP = 0,

(3.25)

∂tCV + V · ∇xCV − DV∆xCV = 0. (3.26)

Coupling conditions to the microscopic problems on I× S0 are

DV∂C V ∂n = ¯cs τs − C V τV , (3.27) V = 0, (3.28) where ¯cs(x , t) = R

Y cs(x , t, y ) dy is the microscopic mean value of cs. Due to

the boundary condition (3.28) the Navier-Stokes system (3.25) decouples from the other equations. Therefore, the velocity field V and the pressure P can be computed in a first step and then, the remaining problem has to be solved for given V and P . To complete the model, consider the boundary conditions

∂CV

∂n = 0, (3.29)

V = 0, (3.30)

on I× (∂Q \ S0) , and initial conditions for x ∈ Q

CV(0, x ) = C_{i ni}V , (3.31)

(43)

3.4 The Two Scale Model 43

• A microscopic Stokes system at every fixed point x ∈ S0 and time t ∈ I

divyv = 0,

−η∆yv +∇yp = 0,

in Ql, (3.33)

with periodic boundary conditions for v with respect to y1, y2. Furthermore, there

are two coupling conditions. On the free boundary Γ this is v = vΓ :=− 1 %V − 1 %E CV τV − cs τs e3. (3.34)

For y3→ ∞, there are the matching conditions

lim y3→∞ ∇yv + (∇yv )> e3= ∇xV|x3=0+ (∇xV ) >_| x3=0 e3, (3.35) lim y3→∞ p = P|x3=0. (3.36)

• A microscopic elastic equation to be solved for every x ∈ S0, t ∈ I

− divyσy(u) = 0, in Qs, (3.37)

This system is completed by a Dirichlet boundary condition

u = b, for y ∈ ˜Γ := Y × {0}, (3.38)

periodic boundary conditions for u with respect to y1, y2, and the coupling

σy(u)n− η ey(v ) n + pn = 0, on Γ, (3.39)

to the Stokes system. Here n is the outer normal vector on Qs at Γ.

• A microscopic phase field model to be solved in I × Y for every x ∈ S0,

τ ξ2∂tφ− ξ2∆yφ + f0(φ) + q(φ, cs, u) = 0, (3.40) ∂tcs+ %s∂tφ− Ds∆ycs = CV τV − cs τs , (3.41)

with Y -periodic initial conditions

cs(0, x , y ) = cs,i ni(x , y ), φ(0, x , y ) = φi ni(x , y ), (3.42)

and periodic boundary conditions with respect to y1, y2. The function f is the

multi-well potential with minima at integer values, e.g. f (φ) =− cos(2πφ), and q(φ, cs, u) = ξRT %s ceqγβ (ceq− cs) g10(φ) + ξhA%s 2ceqγβ σy(u) : ey(u), (3.43)

where the function g₁0 is either g₁0(φ) = 1₂(1− cos(2πφ)) or g0

1(φ) = 1. The first

choice follows [31] and ensures that the corresponding term is only nonzero in the neighborhood of a step, while the second is that of [19].

(44)

The two scale formulation is an alternative approach for solving the model equations numerically compared to direct simulation. The computation of the microstructure has to be done on representative periodicity cells which shrink, from the macroscopic point of view, to single points. The microscopic quantity cs occurs in a coupling term in the

macroscopic equations in an averaged form. As a consequence of that approach it is possible to choose a much coarser grid in the macroscopic domain compared to a direct simulation approach. It is not necessary to resolve the microstructure. The price to pay is, that in every macroscopic grid point on S0one microscopic problem has to be solved.

Since the microscopic problems at different macroscopic points do not influence each other directly, they can be solved in parallel computations. Furthermore, an adaptive strategy as in [42], where only few selected microscopic problems are solved, might be applicable: It requires continuous interscale dependencies, which are proven in sections 5.2.4 and 5.3. This reduces the computation effort significantly.

The above model is a first try to include elastic effects into the model of [10] and [17] without elasticity, for which existence and uniqueness of solutions is proven, the formal derivation of the two scale model is justified rigorously, for both see [17], and numerical experiments are presented with good results, see [10]. This thesis proves existence and uniqueness of solutions for the extended model, but several questions are still open, especially the justification of the homogenization approach, and numerical experiments are still missing.

Also concerning the model, there is room for discussion. It is not clear, for example, how to model the misfit between substrate and layer correctly. In [19], as in most foregone models, this is done as prescribed stress of the form

σ(u)n = b, on Y × {0}, (3.44)

while here, a prescribed displacement is assumed, see condition (3.38). The latter en-sures uniqueness of the solution of the elasticity problem, while a solution for the Neu-mann condition (3.44) in combination with the NeuNeu-mann condition (3.39) and periodic boundary conditions with respect to (y1, y2) can only be unique up to a constant. For

the coupling to the rest of the model, this has no consequences, since only e(u) appears there.

Furthermore, it is questionable, if any prescribed condition is correct at that point, or if rather an interaction between substrate and layer should be allowed, [48]. This would lead to another elasticity problem in the substrate with possibly another free boundary between substrate and layer.

Another point concerns the phase field and its coupling with the elasticity. From the interpretation of φ, it is clear, that its values have to be nonnegative. But that can not be seen from the equation, especially due to the elastic term in (3.43). Furthermore, if the phase field is zero, the elastic energy term in (3.43) should vanish, since there is no solid layer left at that point. Moreover, the growth is modeled to take place at the steps. A possible approach here is to proceed as for the first term in (3.43) and to multiply the elastic term by a function g₂0(φ) which becomes zero for φ _{∈ N}0, for

(45)

3.4 The Two Scale Model 45

example

g₂0(φ) = 1

2(1− cos(2πφ)),

(46)

A two scale model for liquid phase epitaxy with elasticity